let V be RealLinearSpace; :: thesis: for L being Linear_Combination of V holds

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

let L be Linear_Combination of V; :: thesis: ( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

thus L + (ZeroLC V) = L :: thesis: (ZeroLC V) + L = L

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

let L be Linear_Combination of V; :: thesis: ( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

thus L + (ZeroLC V) = L :: thesis: (ZeroLC V) + L = L

proof

hence
(ZeroLC V) + L = L
; :: thesis: verum
let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: (L + (ZeroLC V)) . v = L . v

thus (L + (ZeroLC V)) . v = (L . v) + ((ZeroLC V) . v) by Def10

.= (L . v) + 0 by Th20

.= L . v ; :: thesis: verum

end;thus (L + (ZeroLC V)) . v = (L . v) + ((ZeroLC V) . v) by Def10

.= (L . v) + 0 by Th20

.= L . v ; :: thesis: verum